I think that some of the assumptions here are confused.

Firstly, Aristotle didn't think that everything was made up of four elements. If he thought that, it would be very weird that he actually believed there were more than four elements. (But this doesn't really matter in any case: everyone thinks that things are made up of something, and so we're just quibbling about the number, right?)

Second, you say that it is "comically wrong" to think that heavier things fall faster, but there's a reason why this view hung around for way more than a millennium. Today, physicists still debate this. You should see this article, "Aristotle was Right: Heavier Objects Fall Faster," by two physicists at U Mass Amherst in the European Journal of Physics. But either way, the core idea of Aristotle's view is natural and in many circumstances, the heavier thing really does fall faster. The reason for this is air resistance, and actually, Aristotle developed his view for this very reason: his theory of projectile motion is centered around the role of the medium through which the projectile is being thrown. You can see this popular-science treatment of the debate around whether heavier things fall faster for more, and you'll see why there are some cases in which they do and some where they don't.

The differences between Galileo (whom you are implicitly invoking with the point about heavier objects falling faster) and Aristotle have nothing to do with experimentation in any event. If Galileo and Aristotle both performed an experiment involving a stone in a pendulum, they would have seen different things. Aristotle would have attended to the weight of the stone, the vertical height to which it had been raised, and the time required for it to achieve rest. Aristotle would have ignored radius, angular displacement, and time per swing.

These last were salient to Galileo because he treated pendulum swings as constrained circular motions. The Galilean quantities would be of no interest to an Aristotelian who treats the stone as falling under constraint toward the center of the earth.

As for the bulk of your question: we have no reason to think that Aristotle thought that experimentation was fit for slaves. I don't believe he ever comments on experimentation at all, certainly not to say that only slaves performed experiments. Clearly, Aristotle's science was built on observation first and foremost, or else he never would have made reports such as the following:

As many as (hosa) are four legged and live-bearing all (panta) have an esophagus and windpipe, arranged just as in human beings; it is also arranged similarly in as many of the four legged animals as lay eggs, and in the birds; but they differ in the forms (tois eidesi) of these parts. Generally, all and only (panta hosa) those receiving air by inhaling and exhaling in every case (panta) have a lung, windpipe and esophagus; and the position of the esophagus and windpipe is alike in all, but the organs are not, while lung is neither alike in all nor is it positioned in a like manner. And again as many as (hosa) have blood all (panta) have a heart… . But not all [blooded animals] have a lung, for instance fish do not, nor any other animal there might be that has gills. (History of Animals 505b23–506a12).

The whole History of Animals is just a report of empirical facts about animals, for instance.

But the thought of performing experiments in controlled environments doesn't seem to have occurred to him. To be honest, I don't see what would have been added by experimentation in his science.

Consider the following:

These things, then, have now been said by way of outline to provide a taste of what things need to be studied, and what it is about them that needs to be studied, in order that we may first grasp the differences and the attributes belonging to all animals. After we do this, we must attempt to discover the causes. For it is natural to carry out the investigation in this way, beginning with the inquiry into each thing; for from these inquiries it becomes clear both about which things (peri hôn) the demonstration (tên apodeixin) should be and from which things (ex hôn) it should proceed (HA 491a7–14).

Aristotle identifies what he wants to study: the differences and attributes belonging to animals. What would experimentation in a controlled environment add to merely observing animals? He also evidently doesn't have a problem dissecting animals, which is the only way he could have known about windpipes, etc.

One could easily imagine that Aristotle would have been curious about how the blood of animal, for example, would respond to being cooled, or something like that, and so then he would cool it and notice that it, say, coagulated less efficiently in that case. Does that count as an experiment? If so, then, yes, Aristotle certainly experimented, but the experiment would be reported to us not as such but only as an observation that the blood of animal doesn't coagulate efficiently when cooled. It wasn't so important for him to lay out an empirical methodology, etc. (In many other ways, though, methodology was terribly important to him!) I think the idea is that Aristotle would have seen this as a rather unremarkable form of observation.

(This is actually a rather uncontroversial claim. The early modern scientist Bacon in his Novum Organum (1620) did criticize Aristotle's experiments for often not being sufficiently good, but he certainly recognized that there were 'references' to experiments in his works. They just are not references to experiments as such: e.g., something like 'upon experimentation, it was discovered that...'. Instead, he just reports the conclusions.)

About 25% of the surviving pages of Aristotle's works are dedicated to zoology, and the project that he and his associates undertook to gather all the data he used to develop his biology is breath-takingly large. Nothing like it was undertaken again in Europe until the 16th century. That's why even in the 19th century, the famous astronomist Richard Owen said that "Zoological Science sprang from his [that is, Aristotle’s] labours, we may almost say, like Minerva from the Head of Jove, in a state of noble and splendid maturity." I think in this light, he should be shown some goodwill! No, he didn't get everything right, and there are some real embarrassments in ancient Greek science, but even the things that he got wrong were actually less wrong than you might think, and I believe the reason for that is that he cleaved very closely to observation-making, even if he didn't approximate the modern-day form of experimenting in controlled environments, which is a subject on which he says nothing in particular.

the explanation I've always heard about this is that the greeks thought that experiments were manual labor, and manual labor was for slaves, hence they could only limit themselves to thinking

However that explanation seems... way too simplistic, it seems that there should be more to this

Asking about experimentation in the rest of ancient Greek science is way too big for a Reddit post, but yes, there were experiments throughout ancient Greek science. But I can give some examples without much difficulty. In the Hippocratic Corpus there are several texts which refer to simple investigations of air and water pressure, as for example the creation of what we should call a partial vacuum by inverting a narrow-necked vessel containing water or oil, and the release of the vacuum by piercing a hole in the vessel. But two rather more elaborate tests are worth mentioning particularly. One writer describes a test which demonstrates that water finds its own level: this involves setting up an apparatus of three or more intercommunicating vessels on level ground (this point is stressed), and the writer describes how the whole system may be filled or emptied by filling or emptying any jone of the vessels (though he then uses this piece of information to support a highly speculative theory of the interplay of the humours in the body). And another quite ambitious test is described in the treatise On the Nature of the Child (ch. 17, L. VII, 498, i7ff.) This involves putting three different substances (earth, sand and lead-filings) into a bladder full of water and agitating them by blowing on them through a tube let down into the vessel. The author refers to this test to support his theory that the various parts of the body are formed by the action of like coming to like, but it seems probable that this test was originally designed not to illustrate the action of like-to-like, but to study the reactions of substances of different specific gravities in partial suspension in water, and that the author of On the Nature of the Child attempted to adapt this experiment to suit his own purposes. Perhaps the best known example in the Hippocratic Corpus is the test described in On Airs, Waters, Places (chapter 8) in which a bowl of water is left out of doors to freeze and when the water is thawed it is found on being remeasured to be less than the original quantity (a test which the writer supposes supports his contention that freezing causes the 'lightest and finest' part of the water to be dried up and disappear). In Aristotle we find, for example, two references to the fact that if a vessel is heated and then inverted over water, as the air in the vessel cools it contracts and some of the water is drawn up into the vessel, though it may be thought unlikely that Aristotle himself originated this test.